10.4 Inverse Functions.
A. Local Invertibility.
1. Example. Let 𝑓 𝑥 = 𝑥 2 . For any 𝑥0 ∈ (0, ∞)
there is an open interval around 𝑥0 such that 𝑓
restricted to that interval is injective. Similarly
for any 𝑥0 ∈ (−∞, 0). However 𝑓 is not injective
on any open interval containing 0.
2. Remark. Suppose 𝑓: (𝑎, 𝑏) → ℝ is 𝐶 1 on (𝑎, 𝑏)
and that for some 𝑐 ∈ (𝑎, 𝑏), 𝑓 ′ 𝑐 ≠ 0. Then
the following conclusions hold:
a. 𝑓 ′ 𝑥 ≠ 0 on some interval (𝑐 − 𝛿, 𝑐 + 𝛿).
b. 𝑓(𝑥) is monotone on this interval, hence
invertible.
c. 𝑓 −1 is also 𝐶 1 on some open interval around
𝑓 𝑐 , and 𝑓 −1 ′ 𝑦 = 1/𝑓 ′ (𝑓 −1 𝑦 ) for 𝑦 in
this interval.
3. Example. 𝑓: 𝔼2 → 𝔼2 given by
𝑓 𝑥, 𝑦 = (𝑥 2 + 𝑦 2 , 𝑥 + 𝑦)
4. Definition. (Local invertibility) A function
𝕗: 𝔼𝑛 → 𝔼𝑛 is locally one-to-one in an open set
𝑉 if for every 𝕩0 ∈ 𝑉, there is an 𝜖 > 0 such
that 𝕗 restricted to 𝐵(𝕩0 , 𝜖) is one-to-one. If 𝕗 is
one-to-one on a set 𝐸 then we say 𝕗 is globally
one-to-one on 𝐸.
5. Example. Let 𝑓 𝑥, 𝑦 = (𝑥 cos 𝑦 , 𝑥 sin 𝑦) be
defined on the open set 𝑉 = { 𝑥, 𝑦 : 𝑥 > 0}.
Then 𝑓 is locally 1 − 1 on 𝑉 but not globally
1 − 1 on 𝑉.
B. The Jacobian.
1. Definition. Suppose that 𝕗: 𝐷 ⊆ 𝔼𝑛 → 𝔼𝑛 is in
𝐶 1 (𝐷, 𝔼𝑛 ). Then the Jacobian of 𝕗 at 𝕩 ∈ 𝐷 is
given by det⁡
(𝕗′ 𝕩 ).
2. Theorem. Suppose that 𝕗: 𝐷 ⊆ 𝔼𝑛 → 𝔼𝑛 , 𝐷 an
open subset of 𝔼𝑛 , is in 𝐶 1 (𝐷, 𝔼𝑛 ), and suppose
that det⁡
(𝕗′ 𝕩 ) ≠ 0 for all 𝕩 ∈ 𝐷. Then 𝕗 is
locally one-to-one in 𝐷.
C. Inverse Function Theorem.
1. Lemma. (Open Mapping Theorem, Thm.
10.4.2). Suppose 𝕗 ∈ 𝐶 1 (𝐷, 𝔼𝑛 ) where 𝐷 ⊆ 𝔼𝑛
is open. If det⁡
(𝕗′ 𝕩 ) ≠ 0 for all 𝕩 ∈ 𝐷, then 𝕗 is
an open mapping, that is, 𝕗 maps open subsets
of 𝐷 to open subsets of 𝔼𝑛 .
2. Theorem (10.4.3). Suppose 𝕗 ∈ 𝐶 1 (𝐷, 𝔼𝑛 )
where 𝐷 ⊆ 𝔼𝑛 is open. If det⁡
(𝕗′ 𝕩 ) ≠ 0 for all
𝕩 ∈ 𝐷, and if 𝕗 is globally one-to-one on 𝐷, then
𝕗−1 ∈ 𝐶 1 (𝑓 𝐷 , 𝔼𝑛 ) and
𝕗−1 ′ 𝕗 𝕩 = (𝕗′ 𝑥 )−1